Earth has about five million times as much mass, so the Sun cutout would have about a fifth of a millionth Earth's gravitational pull.
EDIT: In the grand spirit of what-if.xkcd.com I think it behooves us to take a humorously dark scientific look at this.
A larger problem would be 1.14×10¹⁸ kg of cardboard suddenly falling onto Earth's surface. Aside from any effects from it, like, directly impacting, you've just dumped 1.14×10¹⁸ kg of wood pulp onto Earth. Aside from any unpleasant effects from chemical additives, or blotting out the Sun's light to plants and causing biological collapse, I imagine that there could be some other unpleasant effects:
Earth's atmosphere is mostly nitrogen and oxygen, and by volume about 20% oxygen, so we'll back-of-the-napkin it and say that about 20% of the mass is oxygen. So about 1×10¹⁸ kg of oxygen, fairly close to the mass of the cardboard cutout.
That cardboard is presumably flammable.
So if this vast expanse of cardboard ignites --- like, from heat produced by falling through Earth's atmosphere, falling on any open flame in the world, or whatever, but seems like a pretty safe bet that something will touch it off --- I'd assume that a considerable portion of Earth's oxygen supply would be converted to water and carbon dioxide in the resulting combustion reaction. Even aside from the global wildfire itself, that seems like it'd be pretty bad news for humans.
The Sun's radius is ~696,000,000 meters, so the surface area of a perfectly circular cutout would be 1.5218e18 square meters. An article I found says that cardboard used for packing is about 0.35-0.4 kg per square meter, so taking an average of 0.375kg/m^2 gives a total of 5.7069e17kg. This is about the same mass as 40% of all water on Earth.